Q. 22. Find the equation of a plane which contains a given line, and is also perpendicular to a given plane.

OA, OB, OC are three edges of a cube, and OD is the diagonal through O. Show that the projections upon a plane perpendicular to OD of the six edges CB' BB, BD' D'A, ÀA', Â'C, which do not meet the diagonal OD, will form a regular hexagon.

Q. 23. Indicate on a diagram the form of the surface

What must be the value of p in order that the plane

Show that there are two tangent planes parallel to the plane

and give the co-ordinates of their points of contact with the surface.

Show also that the line joining these points of contact passes through the origin.

In Solid Geometry Q.'s 21 and 23 were often well answered; Q. 22 less often.

Q. 24. When is a function f (x) said to have a maximum value? Find the condition that f(x) should have a maximum value when x= a, when the first of the successive differential coefficients of f(x) which does not vanish when x = a is the nth.

A portion of a paraboloid of revolution cut off by a plane perpendicular to the axis has a coaxial right cylinder inscribed in it. Show that the greatest possible volume of this cylinder is the volume of the paraboloid.

Q. 25. Show that the curvature of a circle is measured by the reciprocal of the radius. Established one of the two formulæ

Find p at the points where the axis of a meets the curve

Q. 26. In the curve r = f(e) find an expression for the angle contained by the radius vector and the tangent at any point P on the curve.

Trace the curve r = a sec ± a tan and draw the asymptote.

If a radius vector OPP' be drawn cutting the curve in P and
P', and if the tangents at P and P' meet at 7, prove that
PT
= PT.

In the Differential Calculus Q.'s 24, 25, 26 were all frequently made out quite rightly. Points of weakness were:-In Q. 24 many did not see clearly the essential points that n must be an even number, and f(")(a) negative.

In the latter part of Q. 25 many could not overcome the difficulty involved in finding p at a point where is infinite, or has more than one dx

value.

it is often convenient to change the variable. Explain how to find the proper limits for the new variable: illustrate the method with

Q. 29. The co-ordinates of any point on a cycloid being given in the form

Show that the whole length of a cycloidal are from cusp to cusp is 8a; and that the whole area bounded by this arc and the line on which the generating circle rolls is 3πа2.

Both in the Integral Calculus and in Differential Equations there were many very good answers, though the last part of Q. 27 was rarely established; and in Q. 29 many failed to see that the origin is at the vertex and the axis of x is the tangent there: thus the limits of were not correctly determined, and the wrong area was calculated.

STAGE 7.

Results 1st Class, -; 2nd Class, ; Failed, 1; Total, 1. There was only one candidate; he answered one question fully and two other questions partially.

HONOURS IN DIVISION II.

Results 1st Class,; 2nd Class, 1; Failed, 2; Total, 3.

Three candidates took this paper; the first answered fairly well, the second moderately; the third was disqualified.

DAY EXAMINATIONS.
STAGE 1.

Results 1st Class, 398; 2nd Class, 542; Failed, 208; Total, 1,148.

On the whole the work seems to be better than the work sent up in the Evening examination. It is, perhaps, a little better than the work sent up in the Day examination of last year.

In ARITHMETIC the questions most frequently taken were the following :

Q. 1. Reduce to their simplest forms :

(a) 2% +1%-3;

(b) (91 × 5) ÷ (6} × 1!4).

Find how many times the excess of the greater over the less of the two expressions (a) and (b) is contained in the greater of them.

Q. 4. A merchant sells goods to a customer at a profit of 44 per cent., but the customer becomes bankrupt, and pays only 148. 4hd. in the pound. What per cent. does the merchant gain or lose on the whole transaction?

Q. 6. A man gives by his will one-tenth part of his property to each of his three daughters; he gives a sixth part to each of his two younger sons; he makes his eldest son residuary legatee. In consequence, the eldest son comes in for £5,280. Find the sum received by each of the other children.

The other questions were rightly answered, fairly often.

Q. 2. Find by contracted multiplication the product of 314159 × 87342 so as to obtain the product true to four decimal places.

Divide 173205 by 141421, by contracted division, so as to obtain the quotient true to four decimal places.

From the answers to this question it appeared that some of the candidates can use contracted methods of multiplying and dividing decimals. In this connection it may be observed that if, for instance, the question is to write down 2:8876173 true to five places of decimals, very many do not seem to understand that the answer is 2:88762, not 2:88761.

GEOMETRY. The questions in this section were fairly well answered. But it may be well to notice that the object of teaching Geometry is to impart correct notions of lines, angles, triangles, &c., and correct ways of reasoning about them. Neatly drawn diagrams are often a great help in these respects, but they are not a substitute for correct reasoning.

Q. 8. At a given point in a straight line show how to make an angle equal to a given angle.

A and B are two points on level ground, and P is a somewhat distant inaccessible point, on the same ground; you are provided with a few pegs and some rope; show how you could fix a point Q, such that BQ may be parallel to AP.

The answers to this question would have been better, perhaps, in some cases, if the candidates had realized that the point P is inaccessible. For the most part, however, it seemed plain that if they had gone on to the ground with some rope and a few pegs, they would not have been able to lay down the line BQ.

The following, which requires to be stated at full length, calls for notice. The first part of the question was often answered as follows:-Let the given angle be BAC, and D the point in the line DE; with centre 4 and any radius cut AB in P and AC in Q; also with centre D and the same radius, draw an arc cutting DE in F; with centre F and radius equal to the distance PQ cut the arc in G; join GD; then GDE is the required angle. This, of course, follows from Euclid 1-8. The method-which is really the same as Euclid's-is quite satisfactory, provided it is distinctly understood that the reasoning depends on the equality of the chords FG and PQ. Very commonly, however, it happened that the proof broke down because of a confusion between the chords and the arcs.

Q. 11. Show how to describe a square on a given straight line, having only ruler and compasses. Describe a square, one of whose sides (AB) is two inches long.

Show how to cut off from the square a third part of its area by means of a line drawn through A.

The reasoning in the first part was often unsatisfactory, apparently because the students had not distinctly in mind what they meant by the word " square." The teachers ought to keep the definitions distinctly before their class.

In ALGEBRA the questions most commonly attempted were the following:

--

Q. 13. (a) Find the sum of 3.x + 5y - Gry, 7x - 8y + 54xy, and 11x + 4y - 51xy.

(b) If x =

1, and y = , find the numerical values of the three expressions and of the sum that you have obtained, and show that the sum of the first three values equals the fourth value.

Q. 16. (a) Simplify the following expression:

(26 - a) (a - b) a b 26 a

(b) Find the value of x, for which the following expression equals 0:

where a, b, c denote three different numbers.

Q. 17. Solve the following equations :

In (c) verify your result, by substituting the values of and y, which you have found, in both equations.

Q. 18. An express train, which travels one-third as fast again as an ordinary train, performs a journey of 252 miles in 1 hours less time than the ordinary train. Find the average speed of each train in miles per hour.

In very many cases the results were correct, but it may be noticed that the substitutions in Q. 13 were often omitted, or wrong.

(b) Find the value of c that will make 2x+ x3 + 10x2
divisible by 2x + x + 5 without a remainder.

Q. 15. (a) Write down x2- 7x + 10 in factors, and, assuming that x stands for a positive number, find under what circumstances the given expression will be negative.

(b) Verify your result by considering two cases, viz., first when x stands for 4, secondly, when x stands for 1.

These two questions were taken fairly often, but the answers to Q. 14 (b) and Q. 15 (a) illustrate the difficulty of inducing learners to reason about their results. In the former, they would often find the remainder - c - 15, but could not draw the conclusion, that c must equal 15, if there is to be no remainder. In the second case, they would find that the expression equals (x-5) (x-2), but were quite unable to make out that it is negative only when x is less than 5 and greater than 2.

STAGE 2.

с

Results 1st Class, 306; 2nd Class, 1,167; Failed, 487; Total, 1,960. The work is distinctly better than the work sent up in the Evening examination, and also than that sent up in the Day examination of last

year.

GEOMETRY. The work was mostly directed to Q.'s 21, 22, 23, but there were a good many answers to Q.'s 24, 25, 26.

Q. 21. Show how to divide a given straight line into two parts, so that the rectangle contained by the whole line and one of the parts may be equal to the square on the other part.

If A, B, C be three given straight lines, show how to construct a line X, such that the square on X shall equal the excess of the square on A above the rectangle contained by B and C.

In the deduction, when the side (P) of a square equal to the rectangle under B and C had been found, there ought to have been no difficulty in finding the side of a square equal to the excess of the square on 4 over the square on P, by Euclid I-47. In many cases, however, the method was to draw the square on A, to cut out of it a square equal to the square on P, and then to find a square equal to the remainder by Euclid I-45 and II—14. Of course, such a clumsy method was in most cases ill carried out.

Q. 22. If a straight line drawn from the centre of a circle bisects a chord which does not pass through the centre, show that it cuts the chord at right angles.

Also state and prove the converse of the above theorem.

Let O be the centre of a given circle in which a chord AB is drawn and produced to C so that BC is equal to the radius OA; let CDE be drawn to pass through O and to meet the circumference in D and E. Show that the angle ACO is one-third of the angle AOE.

In the second part the reasoning not unfrequently ran thus :-"In the triangles ADB, ADC the side AB equals the side AC, and the side AD is common, also the angles at D are right angles, therefore the triangles are equal in all respect (Euclid I-26)." The fallacy, whether expressed or implied, that the conclusion follows from Euclid I-26, is one that the teachers ought to be very much alive to.

Q. 23. Two circles touch one another internally, show that the line which joins their centres will, if produced, pass through the point of contact.

Two circles touch one another internally, and the diameter of the smaller circle is greater than the radius of the larger circle. Chords of the larger circle are drawn to touch the smaller circle; show that the longest of these chords is at right angles to the line which passes through the centres of the two circles.

In several cases candidates tried to prove this easy proposition by means of "limits." It is hardly necessary to say that those who are learning the elementary properties of the circle have not reached a point at which they can be trusted to use limits. The result of introducing learners prematurely to such a method is to muddle, not to enlarge, their minds.

Q. 24. Show how to inscribe a circle in the smaller sector formed by joining the centre C' of a given circle with two points, A and B, on its circumference.

If from any point P on the arc of the sector, perpendiculars PM, PN be let fall on the radii CA, CB, show that the diameter of the circle drawn through P, M, N, and the chord MN, are each of constant length.

The first part was answered as often as could be fairly expected; the second part very seldom.

Q. 25. Two given circles touch each other internally; show how to draw a chord of the larger circle, which shall touch the smaller circle and be of a given length.

When the diameters of the given circles are 4in. and 2in. respectively, construct a chord of the larger circle, which shall be 2 in. long and shall touch the smaller circle.

This question was answered oftener than might have been expected.

9291.

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